推 monotones:這是 LaTex語法嗎？ 07/07 07:10
推 micool:u got it. 07/07 07:56
課程名稱︰代數二
課程性質︰數學系選修 數學系碩士班數學組必選
課程教師︰朱樺
開課系所︰數學系所
考試時間︰950624
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(1) (20%)
(a) Let $F$ be a field and $K$ be a Galois extension of $F$. Let $f(x)$
be an irreducible polynomial in $F[x]$. If $g(x)$ and $h(x)$ are
irreducible factors of $f$ in $K[x]$, show that there exists an
automorphism $\sigma$ of $K$ over $F$ such that $\sigma(g)=h$.
(b) Give an example when this conclusion is not valid if $K$ is not
Galois over $F$.
(2) (20%) Let $f(x)=x^4+ax^2+bx+c$ be a quartic polynomial over $\mathbb{R}$,
and let $D$ be the discriminant of $f$. Prove the following statements:
(a) $f$ has four real roots if and only if $D>0$, $a<0$, and $a^2-4c>0$.
(b) $f$ has four nonreal roots if and only if $D>0$, and either
$a \geq 0$, or $a^2-4c \leq 0$.
(c) $f$ has two real and two nonreal roots if and only if $D<0$.
(d) $f$ has at most three roots if and only if $D=0$.
(3) (20%)
(a) Let $A$ be the set of all real number $\alpha$ such that
$\cos \alpha$ is constructible by ruler and compass. Show that $A$
is an additive subgroup of $\mathbb{R}$, and if $\alpha \in A$,
then $\frac{1}{2}\alpha \in A$.
(b) Show that $\cos \frac{\pi}{5}$ is constructible.
(c) Determine all integers $n$ such that the angle of degree $n$ is
constructible.
(4) (20%) Let $K=\mathbb{C}(x)$ and let $\sigma$, $\tau$ be
$\mathbb{C}$-automorphisms defined by
$\sigma: x \mapsto \frac{\sqrt{3}ix+1}{x+\sqrt{3}i}$,
$\tau: \x \mapsto -x$. Let $G$ be the group generated by $\sigma$ and
$\tau$.
(a) Determine the group $G$.
(b) Find the fixed subfield $K^G$ of $G$.
(5) (20%) Let $K=\mathbb{Q}(\sqrt{2}, \sqrt{3},u)$,
where $u^2=(9-5\sqrt{3})(2-\sqrt{2})$.
(a) Show that $K$ is a Galois extension of $\mathbb{Q}$.
(b) Determine the Galois group of $K$ over $\mathbb{Q}$.
(c) Determine the subgroups of $\Gal(K/\mathbb{Q})$ and the
corresponding subfields in the Galois pairing.
(6) (20%)
(a) Find the character table of the quaternion group
$Q=\{\pm 1, \pm i, \pm j, \pm k\}$.
(b) Find all irreducible representations of the group $Q$.
--
╓╮★╓╮ ╓╮
╟╢╰╫╫╮╟╢╓╖╮╮╭╥─╮╓╮ ╮╭╥─╮╭╥─╮
╟╢ ╟╢ ╟╢╟╢║║╟╢ ║╟╢ ║╰╨╥╮╟╢ ║
╟╢ ╟╢ ╟╢╟╢║║╟╢ ║╟╢ ║ ╟╢╟╫─╯
╰╜ ╰╨╯╰╜╰╜╜╜╰╨─╯╰╨─╰╰─╨╯╰╨─╯
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.50.48